# L11n117

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n117 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^5+2 t(1) t(2)^4-4 t(2)^4-3 t(1) t(2)^3+5 t(2)^3+5 t(1) t(2)^2-3 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $z^3 a^7+z a^7-a^7 z^{-1} -z^5 a^5-z^3 a^5+3 z a^5+4 a^5 z^{-1} -2 z^5 a^3-7 z^3 a^3-9 z a^3-4 a^3 z^{-1} +2 z^3 a+3 z a+a z^{-1}$ (db) Kauffman polynomial $-z^7 a^9+4 z^5 a^9-5 z^3 a^9+2 z a^9-3 z^8 a^8+13 z^6 a^8-17 z^4 a^8+6 z^2 a^8+a^8-2 z^9 a^7+3 z^7 a^7+10 z^5 a^7-16 z^3 a^7+5 z a^7-a^7 z^{-1} -9 z^8 a^6+32 z^6 a^6-28 z^4 a^6+3 z^2 a^6+4 a^6-2 z^9 a^5-4 z^7 a^5+30 z^5 a^5-31 z^3 a^5+13 z a^5-4 a^5 z^{-1} -6 z^8 a^4+14 z^6 a^4-2 z^4 a^4-10 z^2 a^4+7 a^4-8 z^7 a^3+23 z^5 a^3-25 z^3 a^3+14 z a^3-4 a^3 z^{-1} -5 z^6 a^2+9 z^4 a^2-10 z^2 a^2+4 a^2-z^5 a-5 z^3 a+4 z a-a z^{-1} -3 z^2+1$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-101χ
2         2-2
0        3 3
-2       53 -2
-4      52  3
-6     45   1
-8    65    1
-10   35     2
-12  25      -3
-14 13       2
-16 2        -2
-181         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.